where \(f_i^k(\mathbf {x},t)\)  is the \(i^{th}\)  discrete probability distribution function for the \(k^{th}\)  component, giving the probability of finding a fluid particle at position \(\mathbf {x}\) , time \(t\)  and with discrete velocity \(\mathbf {c}_i\) . The index \(k\)  is such that \(k=1,2\) , while \(i\)  belongs to the range \(0 \le i \le N_{set}\) , being \(N_{set}\)  the dimension of the set of probability distribution functions. This is equal to 8 for the two-dimensional nine speeds lattice (D2Q9) adopted in this paper.
The time step \(\Delta t\)  is expressed in lattice units [1]} and set to 1, a usual choice in the LBM [2]}. Finally, \(F_i^{rep}\)  is a force aimed at upscaling the repulsive near-contact interactions acting on scales much smaller than the resolved ones [3]}.
